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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Finite-dimensional left ideals in some algebras
associated with a locally compact group


Author: M. Filali
Journal: Proc. Amer. Math. Soc. 127 (1999), 2325-2333
MSC (1991): Primary 43A10, 22D15
Published electronically: April 9, 1999
MathSciNet review: 1487366
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $G$ be a locally compact group, let $L^{1}(G)$ be its group algebra, let $M(G)$ be its usual measure algebra, let $L^{1}(G)^{**}$ be the second dual of $L^{1}(G)$ with an Arens product, and let $LUC(G)^{*}$ be the conjugate of the space $LUC(G)$ of bounded, left uniformly continuous, complex-valued functions on $G$ with an Arens-type product. We find all the finite-dimensional left ideals of these algebras. We deduce that such ideals exist in $L^{1}(G)$ and $M(G)$ if and only if $G$ is compact, and in $L^{1}(G)^{**}$ (except those generated by right annihilators of $L^{1}(G)^{**}$) and $LUC(G)^{*}$ if and only if $G$ is amenable.


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Additional Information

M. Filali
Affiliation: Department of Mathematical Sciences, University of Oulu, SF 90570 Finland
Email: mfilali@cc.oulu.fi

DOI: http://dx.doi.org/10.1090/S0002-9939-99-04793-0
PII: S 0002-9939(99)04793-0
Keywords: Locally compact group, Arens product, representation, amenable, $U$-invariant, finite-dimensional left ideal
Received by editor(s): October 28, 1997
Published electronically: April 9, 1999
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1999 American Mathematical Society